Phase Space Of Solutions Research Articles

On black hole temperature in Horndeski gravity

It has been observed that for black holes in certain family of Horndeski gravity theories Wald's entropy formula does not lead to the correct first law for black hole thermodynamics. For this family of Horndeski theories speeds of propagation of gravitons and photons are in general different and gravitons move on an effective metric different than the one seen by photons. We show that the temperature of the black hole should be modified from surface gravity over 2π to include effects of this effective metric. The modified temperature, with the entropy unambiguously computed by the solution phase space method, yields the correct first law. Our results have far reaching implications for the Hawking radiation and species problem, going beyond the Horndeski theories.

Entropies and the first laws of black hole thermodynamics in Einstein-aether-Maxwell theory

Using the solution phase space method, we investigate the thermodynamics of black holes in Einstein-aether-Maxwell theory, for which the traditional Wald method (covariant phase space method) fails. We show the first laws of thermodynamics and definitive entropy expressions at both Killing and universal horizons for some examples of exact black hole solutions, including three-dimensional static charged quasi-BTZ black hole, two four-dimensional static charged black holes and three-dimensional rotating solution. At Killing horizons the entropies are exactly one quarter of the horizon area, but at universal horizons of three-dimensional black holes, the entropies have a corrected term in addition to the one proportional to the horizon area.

Symmetric and uniform analytic solutions in phase space for Navier–Stokes equations

Symmetric and uniform analytic solutions in phase space for Navier–Stokes equations

Research of a three-dimensional nonlinear dynamic system describing the process of two-level assimilation

The work proposes a new general nonlinear mathematical model describing the social process of two-level assimilation taking into account quadratic members of self-restriction of population growth of three sides. In the case of constant coefficients of the model, the first integral of a three-dimensional dynamic system has been found, which in the phase space of solutions is a cone. The three-dimensional dynamic system is reduced to two-dimensional and with the help of the Bendixon’s criterion the theorem of existence in the first quarter of the phase plane of the closed integral trajectory is proved. Thus, conditions on model parameters are found that do not fully assimilate the third side.

Cosmic branes and asymptotic structure

Superrotations of asymptotically flat spacetimes in four dimensions can be interpreted in terms of including cosmic strings within the phase space of allowed solutions. In this paper we explore the implications of the inclusion of cosmic branes on the asymptotic structure of vacuum spacetimes in dimension d > 4. We first show that only cosmic (d − 3)- branes are Riemann flat in the neighbourhood of the brane, and therefore only branes of such dimension passing through the celestial sphere can respect asymptotic local flatness. We derive the asymptotically locally flat boundary conditions associated with including cosmic branes in the phase space of solutions. We find the asymptotic expansion of vacuum spacetimes in d = 5 with such boundary conditions; the expansion is polyhomogenous, with logarithmic terms arising at subleading orders in the expansion. The asymptotically locally flat boundary conditions identified here are associated with an extended asymptotic symmetry group, which may be relevant to soft scattering theorems and memory effects.

Consideration of Socium as Mechanical System with Stream Control

Mathematical modeling of socium as mechanical system with the material components determined by parts of system with qualitatively excellent complexes of properties concerning production, storage and consumption of production is considered. Existence of objectively operating feedback in the representing system is established. The mathematical model formed in work appears the nonlinear system of the ordinary differential equations and is far-reaching expansion of classical models Lotka and Volterra. The mathematical analysis shows existence of a fixed point in phase space of solutions of system and gives for her analytical expressions. Results of private numerical modeling of the offered model are presented. Results of modeling show manifestations of difficult oscillatory dynamics of phase trajectories of the solution of system and serve as an essential argument of an explanation of real oscillatory processes in the history of social systems.

Ray effect mitigation for the discrete ordinates method through quadrature rotation

Solving the radiation transport equation is a challenging task, due to the high dimensionality of the solution's phase space. The commonly used discrete ordinates (SN) method suffers from ray effects which result from a break in rotational symmetry from the finite set of directions chosen by SN. The spherical harmonics (PN) equations, on the other hand, preserve rotational symmetry, but can produce negative particle densities. By construction, the discrete ordinates (SN) method ensures non-negative particle densities.In this paper we present a modified version of the SN method, the rotated SN (rSN) method. Compared to SN, we add a rotation and interpolation step for the angular quadrature points and the respective function values after every time step. Thereby, the number of directions on which the solution evolves is effectively increased and ray effects are mitigated. Solution values on rotated ordinates are computed by an interpolation step. Implementation details are provided and in our experiments the rotation and interpolation steps only add 5% to 10% to the runtime of the SN method. We apply the rSN method to the line-source and a lattice test case, both being prone to ray effects. Ray effects are reduced significantly, even for small numbers of quadrature points. The rSN method yields qualitatively similar solutions to the SN method with less than a third of the number of quadrature points, both for the line-source and the lattice problem. The code used to produce our results is freely available and can be downloaded [4].

A New Chaotic Oscillator—Properties, Analog Implementation, and Secure Communication Application

This paper reports a new 3-dimensional autonomous chaotic system with four nonlinearities. The system is studied with respect to its numerical solutions in phase space, including sensitive dependence on initial conditions, equilibrium points, bifurcation, and maximal Lyapunov exponent. It is shown that the system is dissipative and has a fractional Lyapunov dimension. Besides, a basin of attraction is determined by the Newton–Raphson’s method. To show its practicality, the new system is implemented by means of an analog electronic circuit. Aperiodicity of the experimental signal is verified by means of an improved power spectral density estimator, viz., the Welch’s method. Also, the correlation dimension is estimated from the experimental time series with the result confirming that the responses are deterministic chaos. Finally, an electronic design of a secure communication application is carried out, wherein a nontrivial square wave is modulated by a master chaotic signal. The modulated signal is subsequently recovered by a slave system, and the fast convergence to zero of the information recovery error substantiates the effectiveness of the design.

A multi-criteria optimization approach for HDR prostate brachytherapy: I. Pareto surface approximation

High dose rate (HDR) brachytherapy planning usually involves an iterative process of refining planning objectives until a clinically acceptable plan is produced. The purpose of this two-part study is to improve current planning practice by designing a novel inverse planning algorithm based on multi-criteria optimization (MCO). In the first part, complete Pareto surfaces were approximated and studied for prostate cases.A Pareto surface approximation algorithm was implemented within the framework of Inverse Planning Simulated Annealing. The Pareto surfaces of 140 prostate cases were approximated with the proposed MCO algorithm. For each case, the Pareto surface was represented by automatically generating 300 Pareto optimal plans, and the clinically acceptable region was identified. Thus, 42 000 Pareto optimal plans were created to characterize Pareto surfaces for all the cases. In addition, the relationship between the clinically acceptable region and four anchor plans was studied. As a result, a set of polynomial regression models was extracted to rapidly predict the clinically acceptable region on the Pareto surface based on anchor plans.Pareto surfaces for HDR brachytherapy prostate cases were well characterized in this study. The proposed regression models may help define the most relevant solution phase space.

ОБ ОБОБЩЕННОЙ КРАЕВОЙ ЗАДАЧЕ ДЛЯ ЛИНЕЙНЫХ УРАВНЕНИЙ СОБОЛЕВСКОГО ТИПА НА ГРАФЕ

On the geometric graph, where in addition to the continuity conditions and balance flow, condition of immobility is first introduced into the vertices of the graph, which is converted to a Dirichlet condition when the graph has one edge with two vertices. To solve this problem we first consider the corresponding Sturm–Liouville problem, and the results are then used to solve the Cauchy problem for two linear models, defined on the graph: Hoff equation and Barenblatt–Zheltov–Kochina equation. A feature of the work is the fact that on each edge of the graph given by the equation with different coefficients, which coupled with the introduction of vertices, is fixed for the first time in this problem. Both models relate to Sobolev type equations, the study of which is experiencing an era of its heyday. Reduction of these equations to an abstract Sobolev type equation makes it possible to apply the method of degenerate semigroups of operators. The phase space of solutions is determined by the phase space method, which consists in reducing the singular equation to a regular equation defined on some subspace of the original space. The obtained results of theorems can be used in consideration of inverse problems, optimal control problems, the initial-end and multipoint problems, and also in consideration of stochastic equations for the models set in a geometric graph.

Black holes with toroidal horizons in ( $$d+1$$ d + 1 )-dimensional space-time

We investigate black holes with toroidal horizons in (d +1)-dimensional space-time. Using the solution phase space method, we calculated conserved charges for these black holes before exploring some features of this metric including its entropy and thermodynamic quantities. Another aspect of the study involves obtaining a general exact static interior solution for uncharged black holes with toroidal horizons in (d+1)-dimensional space-time. Finally, an interior solution for charged black holes is obtained.

A Simple Derivation of Tropical Cyclone Ventilation Theory and Its Application to Capped Surface Entropy Fluxes

AbstractIn a recent study, a theory was presented for the dependence of tropical cyclone intensity on the ventilation of dry air by environmental vertical wind shear. This theory was found to successfully capture the statistics of intensity dynamics in the historical record. This theory is rederived here from a simple three-term power budget and extended to analytical solutions for the complete phase space, including the change in storm intensity itself. The derivation is then generalized to the case of a capped surface entropy flux wind speed, including analytical solutions defined relative to both the traditional potential intensity and the capped-flux potential intensity. The results demonstrate that a cap on the surface entropy flux wind speed reduces the potential intensity of the system and effectively amplifies the detrimental effect of ventilation on the tropical cyclone heat engine. However, such a cap does not alter the qualitative structure of the phase-space solution for intensity change phrased relative to the capped-flux potential intensity. Thus, the wind speed dependence of surface entropy fluxes is important for intensity change in real-world storms, though it is not a necessary condition for intensification in general. Indeed, a residual power surplus may remain available to intensify a storm even in the presence of a cap, though intensification may be fully suppressed for sufficiently strong ventilation. This work complements a recent numerical simulation study and provides further evidence that there is no disconnect between extant tropical cyclone theory and the finding in numerical simulations that a storm may intensify in the presence of capped surface entropy fluxes.

Optimizing energy growth as a tool for finding exact coherent structures

We discuss how searching for finite amplitude disturbances of a given energy which maximise their subsequent energy growth after a certain later time $T$ can be used to probe phase space around a reference state and ultimately to find other nearby solutions. The procedure relies on the fact that of all the initial disturbances on a constant-energy hypersphere, the optimisation procedure will naturally select the one which lies nearest to the stable manifold of a nearby solution in phase space if $T$ is large enough. Then, when in its subsequent evolution, the optimal disturbance transiently approaches the new solution, a flow state at this point can be used as an initial guess to converge the solution to machine precision. We illustrate this approach in plane Couette flow by: a) rediscovering the spanwise-localised `snake' solutions of Schneider et al. (2010b); b) probing phase space at very low Reynolds numbers ($< 127.7$) where the constant linear-shear solution is believed to be the global attractor; and finally c) examining how the edge between laminar and turbulent flow evolves when stable stratification kills the turbulent attractor. We also show that the steady snake solution smoothly delocalises as unstable stratification is gradually turned on until it connects (via an intermediary global 3D solution) to 2D Rayleigh-Benard roll solutions.

Revisiting conserved charges in higher curvature gravitational theories

Restricting the covariant gravitational phase spaces to the manifold of parametrized families of solutions, the mass, angular momenta, entropies, and electric charges can be calculated by a single and simple method. In this method, which has been called "solution phase space method," conserved charges are unambiguous and regular. Moreover, assuming the generators of the charges to be exact symmetries, entropies and other conserved charges can be calculated on almost arbitrary surfaces, not necessarily horizons or asymptotics. Hence, the first law of thermodynamics would be a local identity relating the exact symmetries to which the mass, angular momentum, electric charge, and entropy are attributed. In this paper, we apply this powerful method to the $f(R)$ gravitational theories accompanied by the terms quadratic in the Riemann and Ricci tensors. Furthermore, conserved charges and the first law of thermodynamics for some of their black hole solutions are exemplified. The examples include warped AdS$_3$, charged static BTZ, and 3-dimensional $z=3$ Lifshitz black holes.

Conserved charges and first law of thermodynamics for Kerr–de Sitter black holes

Recently, a general method for calculating conserved charges for (black hole) solutions to generally covariant gravitational theories, in any dimensions and with arbitrary asymptotic behaviors has been introduced. Equipped with this method, which can be dubbed as "solution phase space method," we calculate mass and angular momentum for the Kerr-dS black holes. Furthermore, for any choice of horizons, associated entropy and the first law of thermodynamics are derived. Interestingly, according to insensitivity of the analysis to the chosen cosmological constant, the analysis unifies the thermodynamics of rotating stationary black holes in 4 (and other) dimensions with either AdS, flat or dS asymptotics. We extend the analysis to include electric charge, i.e. to the Kerr-Newman-dS black holes.

TU-AB-BRC-05: Creation of a Monte Carlo TrueBeam Model by Reproducing Varian Phase Space Data

Purpose: To create a Varian TrueBeam 6 MV FFF Monte Carlo model using BEAMnrc/EGSnrc that accurately reproduces the Varian representative dataset, followed by tuning the model's source parameters to accurately reproduce in-house measurements. Methods: A BEAMnrc TrueBeam model for 6 MV FFF has been created by modifying a validated 6 MV Varian CL21EX model. Geometric dimensions and materials were adjusted in a trial and error approach to match the fluence and spectra of TrueBeam phase spaces output by the Varian VirtuaLinac. Once the model's phase space matched Varian's counterpart using the default source parameters, it was validated to match 10 × 10 cm2 Varian representative data obtained with the IBA CC13. The source parameters were then tuned to match in-house 5 × 5 cm2 PTW microDiamond measurements. All dose to water simulations included detector models to include the effects of volume averaging and the non-water equivalence of the chamber materials, allowing for more accurate source parameter selection. Results: The Varian phase space spectra and fluence were matched with excellent agreement. The in-house model's PDD agreement with CC13 TrueBeam representative data was within 0.9% local percent difference beyond the first 3 mm. Profile agreement at 10 cm depth was within 0.9% local percent difference and 1.3 mm distance-to-agreement in the central axis and penumbra regions, respectively. Once the source parameters were tuned, PDD agreement with microDiamond measurements was within 0.9% local percent difference beyond 2 mm. The microDiamond profile agreement at 10 cm depth was within 0.6% local percent difference and 0.4 mm distance-to-agreement in the central axis and penumbra regions, respectively. Conclusion: An accurate in-house Monte Carlo model of the Varian TrueBeam was achieved independently of the Varian phase space solution and was tuned to in-house measurements. KO acknowledges partial support by the CREATE Medical Physics Research Training Network grant of the Natural Sciences and Engineering Research Council (Grant number: 432290).

Dyons and dyonic black holes insu(N)Einstein-Yang-Mills theory in anti–de Sitter spacetime

We present new spherically symmetric, dyonic soliton and black hole solutions of the $\mathfrak{su}(N)$ Einstein-Yang-Mills equations in four-dimensional asymptotically anti--de Sitter spacetime. The gauge field has nontrivial electric and magnetic components and is described by $N\ensuremath{-}1$ magnetic gauge field functions and $N\ensuremath{-}1$ electric gauge field functions. We explore the phase space of solutions in detail for $\mathfrak{su}(2)$ and $\mathfrak{su}(3)$ gauge groups. Combinations of the electric gauge field functions are monotonic and have no zeros; in general the magnetic gauge field functions may have zeros. The phase space of solutions is extremely rich, and we find solutions in which the magnetic gauge field functions have more than fifty zeros. Of particular interest are solutions for which the magnetic gauge field functions have no zeros, which exist when the negative cosmological constant has sufficiently large magnitude. We conjecture that at least some of these nodeless solutions may be stable under linear, spherically symmetric, perturbations.

Flying through polytropes.

Dropping objects into a tunnel bored through Earth has been used to visualize simple harmonic motion for many years, and even imagined for use as rapid transport systems. Unlike previous studies that assumed a constant density Earth, here we calculate the fall-through time of polytropes, models of Earth's interior where the pressure varies as a power of the density. This means the fall-through time can be calculated as the central condensation varies from one to large within the family of polytropes. Having a family of models, rather than a single model, helps to explore the properties of planets and stars. Comparing the family of phase space solutions shows that the fall-through time and velocity approach the limit of radial free-fall onto a point mass as the central condensation increases. More condensed models give higher maximum velocities but do not have the right global properties for Earth. The angular distance one can travel along the surface is calculated as a brachistochrone (path of least time) tunnel that is a function of the depth to which the tunnel is bored. We also show that completely degenerate objects, simple models of white dwarf stars supported by completely degenerate electrons, have sizes similar to Earth but their much higher masses mean a much larger gravitational strength and a shorter fall-through time. Numerical integrations of the equations describing polytropes and completely degenerate objects are used to generate the initial models. Analytic solutions and numerical integration of the equations of motion are used to calculate the fall-through time for each model, and numerical integrations with analytic approximations at the boundaries are used to calculate the brachistochrones in the polytropes. Scaling relationships are provided to help use these results in other planets and stars.

Solution phase space and conserved charges: A general formulation for charges associated with exact symmetries

We provide a general formulation for calculating conserved charges for solutions to generally covariant gravitational theories with possibly other internal gauge symmetries, in any dimensions and with generic asymptotic behaviors. These solutions are generically specified by a number of exact (continuous, global) symmetries and some parameters. We define "parametric variations" as field perturbations generated by variations of the solution parameters. Employing the covariant phase space method, we establish that the set of these solutions (up to pure gauge transformations) form a phase space, the \emph{solution phase space}, and that the tangent space of this phase space includes the parametric variations. We then compute conserved charge variations associated with the exact symmetries of the family of solutions, caused by parametric variations. Integrating the charge variations over a path in the solution phase space, we define the conserved charges. In particular, we revisit "black hole entropy as a conserved charge" and the derivation of the first law of black hole thermodynamics. We show that the solution phase space setting enables us to define black hole entropy by an integration over any compact, codminesion-2, smooth spacelike surface encircling the hole, as well as to a natural generalization of Wald and Iyer-Wald analysis to cases involving gauge fields.

A New Nonrelativistic Investigation for Spectra of Heavy Quarkonia with Modified Cornell Potential: Noncommutative Three Dimensional Space and Phase Space Solutions

A New Nonrelativistic Investigation for Spectra of Heavy Quarkonia with Modified Cornell Potential: Noncommutative Three Dimensional Space and Phase Space Solutions